To put it in a very simple way (although many others will probably expand on it) the Catholic Church is the Church founded upon Peter the Rock.
As Jesus said ‘you are Peter, and upon this rock (in Aramaic, which Jesus spoke, the SAME word was used for Peter’s name and for ‘rock’) I will build my church, and the gates of hell will not prevail against it’. Jesus clearly gave Peter an extraordinary mission even among the Apostles, and guaranteed special protection to the Church founded upon him.
Peter alone uniquely was given the Keys to the Kingdom of Heaven, and the commission to ‘feed my sheep, feed my lambs’. It was also Peter alone that Jesus prayed for at the Last Supper - why? So that he might ‘return and strengthen his brothers’. So there’s no question Peter had primacy among the Apostles.
So we have a special mission and special protection guaranteed to the teaching authority of Peter. We also have more general guarantees given to all the Apostles - ‘what you bind on earth will be bound in heaven, what you loose on earth will be loosed in heaven’.
This means that the Apostles could and did have the right and ability to transmit the fullness of their own powers upon their designated successors. In Peter’s case, his successors were (or became, with Linus, Cletus and Clement who followed him) the Bishops of Rome. In the case of the other Apostles, their powers devolved to the Bishops and partially to the priests of the Church. So the Church founded upon Peter became the Church founded upon the Pope/Bishop of Rome.
Now, Jesus also promised guidance by the Holy Spirit which would lead the Church into all truth, as well as that the gates of hell would not prevail against it. This guarantees that the Church as a whole can never officially teach error. Not that every individual pronouncement by Catholic theologians, bishops or priests is immune, nor all statements by Popes except in specific circumstance.
Now the Catholic Church has by far the best claim to have presented a single, consistent, unified doctrine from the time of Christ. Certain disciplines (the form of the Mass, married v unmarried clergy and so on) have changed of course, but these aren’t fundamental points of the faith. Even doctrines which were explicitly stated more recently, such as the Immaculate Conception and the Assumption of Mary and Papal Infallibility, have been around and been believed by most Catholics for many many centuries beforehand.